Call tsolv

ABSTRACT

A METHOD IS DESCRIBED WHICH INCORPORATES SPARSE MATRIX TECHNIQUES FOR EFFICIENTLY EFFECTING THE BACK SUBSTITUTION STEP IN THE SOLVING OF THE LINEAR SYSTEM   FOR VARIOUS VALUES OF T, AND FOR EACH T, FOR VARIOUS VALUES OF X, THE SOLVING OF SUCH LINEAR SYSTEM USUALLY BEING NECESSARY IN THE SOLVING OF NON-LINEAR DYNAMIC SYSTEMS. THE REQUIRED STEPS IN THE SOLVING OF THE LINEAR SYSTEM ARE L/U (LOWER, UPPER) TRIANGULAR FACTORIZATION, FOWARD ELIMINATION AND THE ABOVE-MENTIONED BACK SUBSTITUTION. EQUATION 1 MAY BE RESTATED AS   A(X,T)Y=EN=COL (0, 0, ..., 1)   WHERE   AND UTILIZING 1-2-3 GNSO TECHNIQUE, THE SOLVING OF EQUATIONS 2A AND 2B IMPLIES THE SUCCESSIVE OPERATIONS OF FACTORIZATION   LA/UA=A(X,T)   AND BACK SUBSTITUTION   UAY=EN   THE INVENTIVE METHOD OVERCOME THE PROBLEM WHICH ARISES IN THE SOLUTION OF EQUATION 4 IN THAT THERE ARE REQUIRED AS OUTPUTS FOR EACH VALUE OF X AND T ONLY PARTIAL ANSWERS   YX=$X*(UX-1EN)=$X*$X   SELECTION OF PIVOTS, (3) IT CONVERTS ALGORITHMS FOR MINIMIZING L/U FACTORIZATION OF MATRICES WITH X AND T VARIABILITY TYPE INTO ALGORITHMS WHICH MINIMIZE MULTIPLES FOR THE COMBINED OPERATIONS OF FACTORIZATION AND PARTIAL BACK SUBSTITUTION, I.E., IT ACHIEVES STEPS (1)-(3), (4) IT ACHIEVES STEPS (1)-(3) IN THE CASE OF MORE THAN TWO DISTINCT VARIABILITY TYPES, (5) IT APPLIES AS WRITTEN TO CASES WHERE A IS IRREDUCBLE BUT ACHIEVES STEPS (1)-(3) WITH REDUCIBLE MATRICES   AND THE CORRESPONDING PARTIAL BACK SUBSTITUTION MATRICES UX AND UT, (2) IT MAKES IDENTIFICATION (1) SOLELY FROM THE PROCESS OF GAUSS ELIMINATION WHEREBY THE OPERATION COUNT FOR THE PARTIAL BACK SUBSTITUTION IS ACCOUNTED FOR DURING THE   $X AND $T   ONCE FOR EACH T, WHERE UX AND UT ARE SUBMATRICES OF UA. THE INVENTIVE METHOD PERFORMS THE FOLLOWING FUNCTIONS: (1) IT IDENTIFIES THE &#34;SPARSEST&#34; PARTIAL OUTPUT VECTORS   YT=$T*(UT-1YX)=$T*$T   ONCE FOR EACH X, AND   WHEREIN $X AND $T ARE LOGICAL VECTORS AND &#34;*&#34; REPRESENTS SCALAR MULTIPLICATION. THE PARTIAL ANSWERS FOR THE PROBLEM IN BACK SUBSTITUTION IS THE ASCERTAINING OF THE SPARSEST MATRICES UX AND UT FROM WHICH THERE MAY E OBTAINED YX AND YT UTILIZING THE PARTIAL BACK SUBSTITUTIONS   YT=$T*Y   AND   YX=$X*Y

DEFENSIVE PUBLICATION UNITED STATES PATENT- OFFICE Published at the request or the applicant or owner in accordance with the Notice of Dec. 16, 1969, 869 0.G. 687. The abstracts of Defensive Publication applications are identified by distinctly numbered series and are arranged chronologically. The heading of each abstract indicates the number of pages of specification. including claims and sheets or drawings contained in the application as originally filed. The files of these applications are available to the public for inspection and reproduction may be purchased for 80 cents a sheet.

Defensive Publication applications have not been examined as to the merits of alleged invention. The Patent Oflice makes no assertion as to the novelty of the disclosed subject matter.

PUBLISHED OCTOBER 9, 1973 915 O.Gr. 385

g wm n o (1) for various values of t, and for each t, for various values of x, the solving of such linear system usually being necessary in the solving of non-linear dynamic systems. The required steps in the solving of the linear system are L/ U (lower, upper) triangular factorization, forward elimination and the above-mentioned back substitution. Equation 1 may be restated as A(x,t)y=e =col (O, 0, 1) (2a) where f A: [aal am l (2b) and utilizing 1-2-3 GNSO technique, the solving of Equations 2a and 2b implies the successive operations of factorization LA/UA=A (x,t) and back substitution The inventive method overcomes the problem which arises in the solution of Equation 4 in that there are required as outputs for each value of x and t only partial answers a x*y and yr t"? wherein 0 and 6 are logical vectors and represents scalar multiplication. The partial answers for the y problem in back substitution is the ascertaining of the sparsest matrices UX and UT from which there may be obtained y and y utilizing the partial back substitutions once for each t, where UX and UT are submatrices of The inventive method performs the following functions: (1) It identifies the sparsest partial output vectors and i and the corresponding partial back substitution matrices UX and UT;

(2) It makes identification (l) solely from the proc ess of Gauss elimination whereby the operation count for the partial back substitution is accounted for during the selection of pivots;

TRANSLATE PROBLEM INTO THE FORM A (L!) y I 2 OUTPUT REOIIESTStY AND H OPTIHALLY TAOTORIZE WITH OPTIHAL ORDERING OF RUWS AND COLUMNS OF A'L UA SUFFICIENT FOR OFTIIIALLY COMPUTING y AND y 95mm: cone XSOLII (i AND TSOLV (9 (3) It converts algorithms for minimizing L/ U factorization of matrices with x and t variability type into algorithms which minimize multiples for the combined operations of factorization and partial back substitution, i.e., it achieves steps (1)(3);

(4) It achieves steps (1)-(3) in the case of more than two distinct variability types;

(5) It applies as written to cases where A is irreducible but achieves steps (1)(3) with reducible matrices.

06L 9, 1973 R. K. BRAYTON ET METHOD FOR EFFECTING BACK SUBSTITUTION IN SPARSE MATRIX TECHNIQUES Filed April 5, 1973 PROBLEM DESCRIPTION TRANSLATE PROBLEM mro THE FORM OUTPUT REQUESTSfl AND y A(t,x)y e BORDER PROBLEM MATRIX WITH 0/P VECTOR A y [A 0] y] e 9'1 1 o INITIALIZE t -24 OPTIMALLY FACTORIZE A+=L+U+= [LA Hu o] INITIALIZE m) 26 WITH OPTIMAL ORDERING 0F Rows AND coumus 0F A=LAUA A CALL XSOLV (y -28 l I MSINCE I =QIUA1, m m? -50 yX y'*( g? #0) AND NO Y 9,; =y*(=0)($'#0) ARE NECESSARY AND SUFFICIENT FOR OPTIMALLY CALL TSOLVW COMPUTING y AND y R 56 t=tf? GENERATE CODE XSOLV (SB) No YES AND TSOLV (9 as 40 t=t+At END 

